The goal of this section is to formulate a different categorical approach to characterizing Ehresmann semigroups
from the one described in [1]}.
In fact, our approach can now be seen as a special case of [2]}
although it in fact arose from reading [3]}.
Our goal is to prove a kind of converse of Proposition REF .
To do this, we shall need a class of actions called étale actions
[4]}, [5]} though we prefer the term `supported actions'.
Let \(S\)  be an inverse semigroup and \(X\)  a set.
Let \(p \colon X \rightarrow \mathsf {E}(S)\)  be a function.
A supported action is a left action \(S \times X \rightarrow X\) , denoted by \((s,x) \mapsto s \cdot x\) , such that \(p(x)\cdot x = x\)  and \(p(s \cdot x) = sp(x)s^{-1}\) .
